Ultra-high resolution imaging devices

ABSTRACT

The invention provides an ultra-high numerical aperture imaging device ( 1 ) comprising two rotationally symmetric curved mirrors ( 11   h   , 12   b ), which can be used to achieve very high concentrations of light or other sorts of wave (or other sorts of physical entities that satisfy equivalent “ballistic” equations of motion) or can be used in reverse to form a beam that has a small angle spread, and which can be combined with a further plane, partially transparent mirror (if necessary with additional components to attenuate and rotate the polarisation of the waves involved), to create a device able to achieve a materially better resolution than that implied by the traditional Rayleigh resolution criterion. A detailed method for designing such a device is also disclosed.

The present invention relates to ultra-high resolution imaging devices,including devices using light or other sorts of electromagnetic wavesfor high resolution lithography for, say, semiconductor or microchipmanufacture. The invention involves a combination of an ultra-highnumerical aperture imaging system in conjunction with a suitablystructured mirror and other associated components designed to result inthe boundary conditions to the wave equation arising from the devicemore nearly approximating to that required to generate an implodingdipole solution to Maxwell's equations. The invention also provides anultra-high numerical aperture imaging system using two suitably shapedreflectors, of relatively simple construction which have furtherpotential uses beyond those conventionally relating to ultra-highresolution.

In the context of the present invention “ultra-high resolution” meanshaving a resolving power better than that implied by the Rayleighresolution criterion and “ultra-high numerical aperture” means that therange of angles that rays make when striking the image plane (if thedevice is being used to concentrate light) span a high proportion of thetotal 2π it steradians possible for light falling onto one side of aplane. To create sharp images (at least for small objeds), an opticalsystem needs to be aplanatic. Geometrical optical theory indicates thatsuch a system must have at least two surfaces at which the waves aredeflected, see e.g. Schulz, G. “Higher order aplanatism”, OpticsCommunications, 41, No 5, 315-319 (1982). The invention provides atwo-mirror aplanatic lens arrangement that simultaneously facilitatesultra-high resolution and achieves a very high angle span into a plane.

Attempts to achieve a complete angle span using a combination of amirror and a refracting surface, rather than two mirrors, havepreviously been described by Benitez, P. and Miñano, J. C.Ultrahigh-numerical-aperture imaging concentrator, J. Opt. Soc. Am. A14, No 8, 1988-1997 (1997), and in other papers by the same authors.However, the mirror plus refractor arrangement they describe requiresthe image plane to be embedded within a material with refractive indexgreater than unity, which is considerably less practical than anapproach in which both deflecting surfaces are mirrors.

Benitez and Miñano appear to have developed their ideas from non-imagingsystems of relatively similar layout that were able to achieve very highconcentrations for sources that were not very small. Their imaginglayouts are in effect limiting cases of their non-imaging systems whenthe (far away) source object becomes very small. Other more traditionalforms of non-imaging system are known, such as the Compound ParabolicConcentrator (CPC) described in Welford, W. T. & Winston, R. Highcollection nonimaging optics (Academic Press, 1989). However, thepresent invention differs from these systems in that it is imagingrather than non-imaging and, as is apparent from a cross-section takenthrough an axis of symmetry, a device according to the present inventioncomprises two separate deflecting surfaces not one, as is the case witha CPC. Additionally, in the limit for the CPC as the (far away) sourcebecomes small, the CPC simply becomes arbitrarily long. A deviceaccording to the present invention is readily distinguishable.

Some aplanatic two-mirror arrangements have also been previouslydescribed. These include:

(a) Siemens-Reiniger-Werke Aktiengesellschaft “Improvements in orrelating to optical mirror systems having aspherical surfaces”, in GBPatent No GB 0 717 787 (1952). This patent describes a two-mirroraplanalic device, without explicitly specifying any limitation on thenumerical aperture involved. However, it does not indicate how toachieve an ultra-high numerical aperture, nor do the Figures that itcontains envisage such a device. The patent relates primarily to thedesign of X-ray telescopes which, because of the physical nature ofreflection of X-rays, would not work if the device involved had a veryhigh numerical aperture. Furthermore, despite making reference to atwo-mirror aplanatic device, the Siemens-Reiniger-WerkeAktiengesellschaft patent does not indicate how to define the shape ofthe two mirrors involved.

(b) Mächler, Glück, Sclemmer and Bittner “Objective with asphericsurfaces for imaging microzones”, in U.S. Pat. No. 4,655,555 (1984)concentrates on mirrors that use total internal reflection. It includesreference to a special case of an aplanatic two-mirror arrangementinvolving two confocal equally-sized ellipsoids. It concentrates onother confocal mirror arrangements (as does Hunter “Confocal reflectorsystem” in U.S. Pat. No. 4,357,075 (1980)), although these mirrorlay-outs are not actually aplanatic except in the special case of thetwo confocal equally-sized ellipsoids). However, U.S. Pat. No. 4,655,555also refers to an article by Lawrence Mertz entitled “Geometrical Designfor Aspheric Reflecting Systems”. Applied Optics, 18, pages 4182-4186(1979), which does appear to describe (in its FIG. 10) a very highnumerical aperture aplnatic two-mirror arrangement, again focusing onmicroscopy. Pioneer “Manufacture of reflective type multiple-degreeaspherical optical control system” in Japanese Patent JP 57141613 (1981)refers to an efficient means of producing a two mirror aplanaticarranement using a grip and press work plated by aluminium by vapourdeposition.

(c) Döring in German Patent DE 2916741 notes that such arrangerrents canbe used as optical collectors for solar cells, and the figures suggestreference to aptanatic rather than merely confocal arrangements.

However, none of the above indicate how the precise positioning of themirrors can be identified. The present invention therefore embodies asignificant departure from and advance over the various prior artsystems not only because it refers to ultra-high resolution devices butalso because it provides a simple methodology for identifying theprecise positioning of such aplanatic mirror pairs. In certain preferredembodiments it also incorporates other refinements not described in theabove references.

According to one aspect of the present invention a high numericalaperture imaging device comprises first and second axially-symmetriccurved mirrors for focussing the image of an object onto an image plane,wherein the first and second curved mirrors are arranged to effectivelycreate inwardly imploding dipole-like solutions to the applicable waveequation, to concentrate the light flux arriving at the image plane froma given point in the object more than would be possible were the imageformation to be subject to the diffraction limits that generally applyto far field devices.

In a preferred embodiment, the device further comprises a plane mirror,wherein the plane mirror is partially transparent and is positioned inor closely adjacent to the image plane.

A device according to the invention may further comprise a waveattenuation element and/or wave polarisation-rotating element toattenuate and/or rotate the polarisation of the waves traversing thedevice so that the spatial distribution of the amplitude andpolarisation of a wavefront as it approaches the plane mirror isrendered more closely consistent with that required to generatedipole-like solutions to the wave equation.

Two mirror ultra-high numerical aperture imaging devices according tothe invention may have practical application for several possible uses,including, for example:

(a) use to concentrate sunlight to a high temperature, indeed the secondlaw of thermodynamics indicates that the temperatures reached could bedose to the temperature of the sun's photosphere, i.e. to in excess of4,000° K. At such temperatures, unusual ways of converting sunlight toelectric power (e.g. use of thermionic emission) could be facilitated bya device according to the invention;

(b) as solar concentrators made out of lightweight mirrors (for example,using thin films whose shapes remain stable because of rotation [whichmay require only an initial impetus in a suitable frictionlessenvironment, such as a vacuum, or which could otherwise be achieved witha suitable drive mechanism] or because they are part of an inflateddevice) such that the power to weight ratio of such an apparatus if usedwith a lightweight way of converting sunlight to energy could besufficiently high to permit powered flight (e.g. the sunlight could beused to create direct thrust by evaporation of a solid or liquidpropellant);

(c) to concentrate other types of waves such as sound waves or othersorts of electromagnetic radiation like radio waves (e.g. as analternative to existing parabolic satellite TV dish design);

(d) (when used in reverse) to create narrow beams, e.g. efficient beamformation from light emitted by a light emitting diode in say an opticalnetwork;

(e) to create uffra-high resolution imaging devices probably in tandemwith additional “near field” components;

(f) for concentrating or projecting objects such as gas or dustparticles (the trajectories of objects travelling “allistically” are thesame as light rays, i.e. straight lines until the object bounces off asurface in the same sort of fashion as a light ray bounces off a mirror,thus the same layouts might also be relevant in the context of such“ballistic” materials).

In such examples, an ultra high numerical aperture usually providesadvantages, e.g. in (a) and (b) it makes it possible to approach moreclosely the temperature defining the thermodynamic upper limit, in (c)it improves the quality of the signal received for the same aperturearea, in (d) it reduces the power required for the same usable energyoutput and in (e) it ensures that the required boundary conditions canapproximate those required to generate an imploding dipole solution tothe wave equation. [The “traditional” parabolic dish achieves about ¼ ofthe thermodynamic ideal according to Welford, W. T. & Winston, R. Highcollection nonimaging optics (Academic Press, 1989).]

Alternative embodiments according to the present invention may includeone or more additional focussing mirrors (above two) and/or non-imagingelements for further improving the device to achieve higher orderaplanatism.

The invention further provides a method for designing the curved mirrorsfor use in such high numeric aperture imaging devices.

The various aspects of the invention and the principles underlying itsoperation will now be described in detail and by way of example withreference to the accompanying drawings, in which:

FIG. 1a is a graph of curves for defining the two-mirror surfaces of anembodiment of the invention by rotation about the x-axis;

FIG. 1b is a perspective view in 3-dimensions of the embodiment definedby the curves of FIG. 1a;

FIGS. 2, 3, and 4 are perspective, 3-D views of the mirror surfaces ofalternative embodiments of the invention illustrating the differencesresulting from variation of specified key parameters;

FIG. 5 is a cross-section of a further alternative embodiment of thepresent invention;

FIG. 6 is a cross-section of an embodiment of the present inventionillustrating use for photolithography;

FIG. 7 is an enlarged cross-section of the portion of the image planelabelled “A” in FIG. 6; and

FIG. 8 is an enlarged cross-section of the portion of the object planelabelled “B” in FIG. 6.

In all the figures, for use for high concentration/high resolutionpurposes, the object plane is situated to the left and the image planeis situated to the right.

For a two mirror layout to exhibit first order aplanatism when creatingan image on the plane x=0 and centred on the origin, it is sufficientthat the following conditions are satisfied:

(a) that the device is rotationally symmetric around, say, the x-axis,with the object plane (say at x=f);

(b) that it satisfies the sine criterion; and

(c) that all light rays emanating from the point (f,h,0) in the objectplane that reach the image plane having travelled in the plane z=0 needto go through a single point in the image plane, say (0,ZBh,0), for allsufficiently small h (B being the degree of magnification the deviceproduces and hence independent of h, and Z being ±1 (or a constantmultiple thereof), depending on whether the image is inverted or not).

The reason for sufficiency is that (a) and (c) taken together areequivalent to forming a crisp (aplanatic of order 1) image for raysremaining in a cross-section through the axis, and the furtherimposition of (b) means (for the right sign of Z) that the devicecontinues to form a crisp image even if the rays do not remain whollywithin this cross-section.

These requirements may be employed in an iterative process thatidentifies the positioning of each consecutive point of a cross-sectionthrough the x-axis of each mirror, in the following way. First, it isnecessary to define some suitable notation. For example, suppose thecross-section is in the plane z=0. The x and y coordinates of eachsuccessive point on the two mirrors in this cross-section may be definedto be M₁(t)≡(m_(1,x)(t), m_(1,y)(t),0) (for the mirror nearest theobject) and M₂(t)≡(m_(2,x)(t),m_(2,y)(t),(t),0) (for the mirror nearestthe image), t being an iteration counter. The functionsM₀(t)≡(m_(0,x)(t),m_(0,y)(t),0)≡(f,0,0) andM₃(t)≡(m_(3,x)(t),m_(3,y)(t),0)≡(0,0,0) are set to define the centres ofthe object and image planes respectively. For an imaging device, thesewould be constant and therefore would not vary as t changes (although ifa non-imaging component is introduced as described below then they wouldvary to some extent).

Additionally, a₁(t) and a₂(t) are the angles that tangents to the firstand second mirrors make to the z-axis, and a₀(f) and a₃(t) are theangles that the object and image planes make to the x-axis, i.e. 90° forall t (for the device to be rotationally symmetric about the x-axis),d_(i)(t) (for i=0, 1, 2) is the angle that a ray from M_(i)(t) toM_(i+1)(t) makes to the x-axis and p_(i)(t) (for i=0, 1, 2) is thedistance between M_(i)(f) and M_(i+1)(t). Note that as the light isreflected at each mirror these can be found from the M_(i)(t) asfollows:${d_{i}\quad (t)} = {\arctan \quad \left( \frac{{m_{{i + 1},y}\quad (t)} - {m_{i,y}\quad (t)}}{{m_{{i + 1},x}\quad (t)} - {m_{i,x}\quad (t)}} \right)}$${a_{i}\quad (t)} = {\frac{d_{i}\quad (t)}{2} + \frac{d_{i - 1}\quad (t)}{2}}$${p_{i}\quad (t)} = \sqrt{\left( {{m_{{i + 1},x}\quad (t)} - {m_{i,x}\quad (t)}} \right)^{2} + \left( {{m_{{i + 1},y}\quad (t)} - {m_{i,y}\quad (t)}} \right)^{2}}$

The design process then proceeds iteratively as follows:

(1) Choose suitable parameters to define the size and shape of thedevice. For a far away source (which might be taken to be, say, f=−10⁹,i.e. far away along the negative x-axis) the overall width of the devicewill be defined by the value of B. e.g. if B=b/p₀(0) then the maximumwidth will be b (since the sine criterion implies that this will be thevalue of m_(1,y)(t) corresponding to rays striking the image planetangentially). So the scale of the device can be defined by taking, say,b=1. There are then two further parameters that define the shape of thedevice, which for simplicity can be m_(2,y)(0)=k and m_(2,x)(0)=−q, say.It is possible to then take m_(1,x)(0)=m_(2,x)(0)−δ, where forsimplicity δ is very small (as this then defines one limit of theacceptable range of iterated values).

(2) Assign the following values to m_(1,y)(0) and Z (these ensure thatboth at outset and as the iteration progresses the device satisfies thesine criterion, as long as h is small enough for that purpose):$Z = {{1\quad m_{1,y}\quad (0)} = {- \frac{m_{2,y}\quad (0)}{\left( {{m_{2,x}\quad (0)^{2}} + {m_{2,y}\quad (0)^{2}}} \right)^{1/2}}}}$

(3) Update the values of M₁(t) and M₂(t) using the following iterativeformulae for a suitably small value of h:${{M_{i}\quad \left( {t + 1} \right)} \equiv \begin{pmatrix}{m_{i,x}\quad \left( {t + 1} \right)} \\{m_{i,y}\quad \left( {t + 1} \right)}\end{pmatrix}} = {{M_{i}\quad (t)} + {w_{i}\quad \begin{pmatrix}{\cos \quad \left( {a_{i}\quad (t)} \right)} \\{\sin \quad \left( {a_{i}\quad (t)} \right)}\end{pmatrix}\quad h}}$

 where$w_{2} = {{\frac{p_{1}\quad r_{0}}{p_{0}\quad s_{2}}\quad w_{1}} = {{- {ZB}}\quad \frac{p_{1}\quad s_{3}}{p_{2}\quad r_{1}}}}$

 and r_(i−1)(t)=sin(a_(i−1)(t)−d_(i−1)(t))s_(i)(t)=sin(a_(i)(t)−d_(i−1)(t))

(4) Stop the iteration when m_(2,x)(0) reaches zero (at which pointlight rays striking the image will do so tangentially).

(5) Rotate the curves thus produced, which define the shapes of the twomirrors, around the x-axis to form the complete two-mirror arrangement.

The formulae for w₁(f) and w₂(t) in step (3) can be derived in a varietyof ways. One way is via trigonometry considering three different pathsthat rays must simultaneously be able to travel, which are (i) M₀(0) toM₁(t) to M₂(t) to M₃(0), (ii) M₀(0)+offset to M₁(t) to M₂(t+1) toM₃(0)+Offset and (iii) M₀(0) to M₁(t+1) to M₂(t1) to M₃(0), where“offset” means a point in the object plane and corresponding point inthe image plane a small distance way from the center of the object. Amore general way that can be extended to include more than two mirrorsor deflectors other than mirrors (or with suitable modification tocalculate the degree of aberration arising from a source being otherthan a single point, or to add back in a component of non-imagingbehaviour—see below) is to determine the form of the functionE_(i)(g_(i),g_(i−1)) that represents the distance from M_(i+1)(f) that aray will strike the i+1′th surface if it comes from a point on thei−1′th surface that is a distance g_(i−1)(f) from M_(i−1)(f) via a pointon the 1′th surface that is a distance g_(i) from M_(i)(t), for smallg_(i−1) and g_(i). If deflection occurs by reflection then it can beshown that:${E_{i}\quad \left( {g_{i},g_{i - 1}} \right)} = {{\left( {\frac{r_{i}}{s_{i + 1}} - \frac{p_{i}\quad s_{i}}{p_{i - 1}\quad s_{i + 1}}} \right)\quad g_{i}} + {\left( \frac{p_{i}\quad r_{i - 1}}{p_{i - 1}\quad s_{i + 1}} \right)\quad g_{i - 1}} + {\left( \frac{2p_{i}}{s_{i + 1}} \right)\quad a_{i}^{\prime}\quad g_{i}}}$

where a_(i)′(t)g_(i) is the difference between the angle that the ithmirror makes to the x-axis at point M_(i)(t) and the angle that it makesat a distance g_(i) from M_(i)(t) (for a smooth surface this must be alinear function of g_(i) for small g_(i). a_(i)′(t) being the derivativeof a_(i)(t) with respect to t). Requirement (c) is then equivalent tow₁, w₂, a₁′w₁ and a₂′ w₂ satisfying the following four simultaneousequations:

w ₂ h=E ₁(w ₁ h,0) 0=E ₂(w ₂ h,w ₁ h) w ₂ h=E ₁(0,h) ZBh=E ₂(w ₂ h,0)

(the first two being equivalent to the requirement that rays startingfrom the centre of the object plane eventually strike the image plane atits centre, and the second two being equivalent to the requirement thatrays starting a distance h away from the centre eventually strike theimage plane a distance ZBh from its centre), w₁(t) and w₂(t) musttherefore satisfy the following relationships, which in turn lead to theformulae in step (3) of the iteration:${w_{2}\quad h} = {{E_{1}\quad \left( {0,h} \right)} = {{{\frac{p_{1}\quad r_{0}\quad h}{p_{0}\quad s_{2}}\quad 0} - {ZBh}} = {\left( {{E_{2}\quad \left( {{w_{2}\quad h},{w_{1}\quad h}} \right)} - {E_{2}\quad \left( {{w_{2}\quad h},0} \right)}} \right) = {\frac{p_{2}\quad r_{1}}{p_{1}\quad s_{3}}\quad w_{1}\quad h}}}}$

If the waves were deflected by refraction or diffraction thenE_(i)(g_(i),g_(i−1)) would need to be modified appropriately. To workout the degree of aberration (at least for rays remaining in across-sectional plane) E_(i)(g_(i),g¹⁻¹) is expanded in higher powers ofg_(i−1) and g_(i) whilst still retaining the positioning defined above.For more than two deflecting surfaces then it is still possible to usethe same sort of approach as above but there are then more unknowns thanthere are equations that need to be satisfied. The extra degrees offreedom that this introduces may be used to satisfy the additionalequations required to achieve higher order aplanatism.

For example, considering a device with three or more mirrors or wheredeflection occurs via some other means, and where it is desired toarrange for the mirrors to exhibit higher order aplanatism whilst stillproviding an ultra-high numerical aperture device, the analysis mayproceed as follows. Let the n deflecting surfaces now be M₁(t), M₂(t), .. . , M_(n)(t) where M_(i)(f)=(m_(i,x)(t), m_(i,y)(f)), the object planebeing M₀(t) and the image plane being M_(n+1)(t). As before, consider aray that remains in the xy plane and that starts a distance g_(i−1)along the (i−1)′th mirror from M_(i−1)(f), which strikes the i′th mirrorat a point g_(i) from M_(i)(t) and after deflection there strikes the(i+1)′th mirror at a distance g_(i+1) from M_(i+1)(t). Suppose that theangles the entry and exit rays at a deflecting surface, and the tangentto the deflecting surface make to the x-axis are d_(entry), d_(exit) anda _(junc) respectively. Then d_(exit)=f(a_(junc), d_(entry)) where fdepends on the type of deflection that is occurring. For example, forreflection, refraction and diffraction:

f _(reflection)(a _(junc) ,d _(entry))=2a _(junc) −d _(entry)

f _(refraction)(a _(junc) ,d _(entry))=a _(junc)−π/2+arcsin(N sin(π/2−a_(junc) +d _(entry)))

where $N = \frac{n_{entry}}{n_{exit}}$

 since

n _(exit) sin(π/2−a _(junc) +d _(exit))=n _(entry) sin(π/2−a _(junc) +d_(exit))

 and n_(entry) and n_(exit) are the refractive indices on the relevantsides of the refractor, and${f_{diffraction}\quad \left( {a_{junc},d_{entry}} \right)} = {a_{junc} - {\pi/2} + {\arcsin \quad \left( {\frac{m\quad \lambda}{L} + {\sin \quad \left( {{\pi/2} - a_{junc} + d_{entry}} \right)}} \right)}}$

where L is the distance between successive grating lines in thediffracting surface (assuming that the pattern can be modelled in someperiodic fashion akin to a diffraction grating), and m is an integerrepresenting the ‘order’ of the diffraction, since:

sin(π/2−a _(junc) +d _(exit))=in.λ/L+sin(π/2−a _(junc) +d _(entry))

For simplicity define r_(i) and s_(i) as before and:

X _(i−1)(t)=cos(a _(i−1)(t)−d _(i−1)(t))Y _(i)(t)=cos(a _(i)(t)−d_(i−1)(t))

and drop the parameter t for d_(i)(f) etc., but retain the fullexpression whenever the parameter is different to t. Then, to firstorder, i.e. when g_(i−1), g_(i) and g_(j+1) are small:$d_{entry} = {d_{i - 1} + \frac{{g_{i}\quad s_{i}} - {g_{i - 1}\quad r_{i - 1}}}{p_{i - 1}}}$a_(junc) = a_(i) + a_(i)^(′)  g_(i)$d_{exit} = {d_{i} + \frac{{g_{i + 1}\quad s_{i + 1}} - {r_{i}\quad m_{i}}}{p_{i}}}$

 d _(exit) =f(a _(junc) ,d _(entry)) and d _(i) =f(a _(i) ,d _(i))→g_(i+1) =E _(i)(g _(i) ,g _(i−1))=L _(i,0) g _(i) +L _(i,1) g _(i−1) +H_(i) a _(i) ′g _(i)

where as previously, if deflection occurs by reflection:$L_{i,0} = {{\frac{r_{i}}{s_{i + 1}} - {\frac{p_{i}\quad s_{i}}{p_{i - 1}\quad s_{i + 1}}\quad L_{i,1}}} = {{\frac{p_{i}\quad r_{i - 1}}{p_{i - 1}\quad s_{i + 1}}\quad H_{i}} = \frac{2p_{i}}{s_{i + 1}}}}$

However, it is possible to expand to higher powers of h, e.g. to expandto

g _(i) =c _(i,1) h+c _(i,2) h ² o(h ³).

Then:

c _(i+1,1) =E _(i,1)(c _(i,1) ,c _(i−1,1)) c _(i+1,2) =E _(i,1)(c _(i,2),c _(i−1,2))+E _(i,2)(c _(i+1,1) , c _(i,1) ,c _(i−1,1))

where

E _(i,1)(g _(i+1) ,g _(i) ,g _(i−1))≡E _(i)(g _(i+1) ,g _(i) ,g _(i−1)).

and for deflection by reflection it appears that:${E_{i,2}\quad \left( {g_{i + 1},g_{i},g_{i - 1}} \right)} = {\frac{p_{i}}{s_{i + 1}}\quad \begin{pmatrix}{{a_{i}^{''}\quad g_{i}^{2}} - \frac{{Y_{i + 1}\quad a_{i + 1}^{\prime}\quad g_{i + 1}^{2}} - {X_{i}\quad a_{i}^{\prime}\quad g_{i}^{2}}}{2p_{i}} - \frac{{Y_{i}\quad a_{i}^{\prime}\quad g_{i}^{2}} - {X_{i - 1}\quad a_{i - 1}^{\prime}\quad g_{i - 1}^{2}}}{2p_{i - 1}} +} \\{\frac{\left( {{s_{i + 1}\quad g_{i + 1}} - {r_{i}\quad g_{i}}} \right)\quad \left( {{Y_{i + 1}\quad g_{i + 1}} - {X_{i}\quad g_{i}}} \right)}{p_{i}^{2}} +} \\\frac{\left( {{s_{i}\quad g_{i}} - {r_{i - 1}\quad g_{i - 1}}} \right)\quad \left( {{Y_{i}\quad g_{i}} - {X_{i - 1}\quad g_{i - 1}}} \right)}{p_{i - 1}^{2}}\end{pmatrix}}$

The degree of aberration G(h) arising from the two mirror arrangementsdescribed previously may then be derived as follows:${G\quad (h)} = {\frac{g_{4}}{g_{1}} = {\frac{{ZBh} + {c_{4,2}\quad h^{2}}}{h} = {{ZB} + {c_{4,2}\quad h} + {o\quad \left( h^{2} \right)}}}}$

where

c _(1,1)=1 c _(2,1)=0 c _(3,1) =ZB c _(1,2)=0 c _(2,2)=0

c _(3,2) =E _(2,1)(0,0)+E _(2,2)(c _(3,1) ,c _(2,1) ,c _(1,1))=E_(2,2))(c _(3,1) ,c _(2,1) ,c _(1,1))

and

c _(4,2) =E _(3,1)(c _(3,2),0)+E _(3,2)(c _(4,1) ,c _(3,1) ,c _(2,1))

The term in a″ can probably most easily be found by carrying out twoconsecutive iterations and calculating it from the difference of twoconsecutive values of a′.

If more mirrors are to be added, then by solving progressively moresimultaneous equations it is possible to arrange for the power seriesexpansion of G(h) to be arbitrarily dose to the thermodynamic ideal(which to second order is ZBh+0.h²). For example, suppose a furtherreflector is to be added. There are then six simultaneous equations (tofirst order) that need to be satisfied, including a parameter Q whicharises because it is not obvious at first sight where the off-centrelight will strike the third mirror. They are:w₂  h = E₁  (w₁  h, 0)  w₃  h = E₂  (w₂  h, w₁  h)  0 = E₃  (w₃  h, w₂  h)w₂  h = E₁  (0, h)  Qw₃  h = E₂  (w₂  h, 0)  ZBh = E₃  (Qw₃  h, w₂  h)

In fact, Q is constrained so that (1−Q)=(ZB)/(L_(2,1)L_(4,1)), but eventaking this into account there are still just five unknowns but sixsimultaneous equations. This therefore introduces a degree of freedomwhich can then be varied to ensure that at each iteration G(h) has theform required for second order aplanatism, at least for rays that remainin the xy plane. A similar analysis must be carried out to ensure thatsecond order aplanatism also arises for rays that do not remain withinthe xy plane (the paper by Schulz referred to above suggests that ingeneral this may require a further mirror). The end result is as beforean iterative process that simultaneously defines the location of eachconsecutive point along each mirror (for some suitably small h). Againthis is seeded with parameters that will satisfy the sine criterion (andappropriate values for Z, B and h) and iterated forwards until the rayheading towards the image is tangential to the image plane. The sameapproach can be extended to higher aplanatic orders by introducingfurther terms such as c_(i,3) and further functions E_(i,3) andincorporating further mirors in the iterative process.

The same framework can be used for monochromatic light deflected in arefractive or diffractive manner. If deflection occurs by refractionthen it appears that:$L_{i,0} = {{\frac{r_{i}}{s_{i + 1}} + {\frac{{Ns}_{i}^{2}\quad p_{i}}{r_{i}\quad s_{i + 1}\quad p_{i - 1}}\quad L_{i,1}}} = {{{- \frac{{Ns}_{i}\quad p_{i}\quad r_{i - 1}}{r_{i}\quad s_{i + 1}\quad p_{i - 1}}}\quad H_{i}} = \frac{p_{i}\quad \left( {r_{i} - {N\quad s_{i}}} \right)}{r_{i}\quad s_{i + 1}}}}$${E_{i,2}\quad \left( {g_{i + 1},g_{i},g_{i - 1}} \right)} = {\frac{p_{i}}{r_{i}\quad s_{i + 1}}\quad \begin{pmatrix}{{r_{i}\quad \frac{{Y_{i + 1}\quad a_{i + 1}^{\prime}\quad g_{i + 1}^{2}} - {X_{i}\quad a_{i}^{\prime}\quad g_{i}^{2}}}{2p_{i}}} - {N\quad s_{i}\quad \frac{{Y_{i}\quad a_{i}^{\prime}\quad g_{i}^{2}} - {X_{i - 1}\quad a_{i - 1}^{\prime}\quad g_{i - 1}^{2}}}{2p_{i - 1}}} -} \\{{\frac{\left( {r_{i} - {N\quad s_{i}}} \right)\quad a_{i}^{''}}{2}\quad g_{i}^{2}} - {r_{i}\quad \frac{\left( {{s_{i + 1}\quad g_{i + 1}} - {r_{i}\quad g_{i}}} \right)\quad \left( {{Y_{i + 1}\quad g_{i + 1}} - {X_{i}\quad g_{i}}} \right)}{p_{i}^{2}}} +} \\{N\quad s_{i}\quad \frac{\left( {{s_{i}\quad g_{i}} - {r_{i - 1}\quad g_{i - 1}}} \right)\quad \left( {{Y_{i}\quad g_{i}} - {X_{i - 1}\quad g_{i - 1}}} \right)}{p_{i - 1}^{2}}}\end{pmatrix}}$

whilst for diffraction then it appears that:$L_{i,0} = {{\frac{p_{i}}{r_{i}\quad s_{i + 1}}\quad \left( {\frac{r_{i}^{2}}{p_{i}} + \frac{s_{i}^{2}}{p_{i - 1}}} \right)\quad L_{i,1}} = {{{- \frac{{Ns}_{i}\quad p_{i}\quad r_{i - 1}}{r_{i}\quad s_{i + 1}\quad p_{i - 1}}}\quad H_{i}} = \frac{p_{i}\quad \left( {r_{i} - s_{i}} \right)}{r_{i}\quad s_{i + 1}}}}$${E_{i,2}\quad \left( {g_{i + 1},g_{i},g_{i - 1}} \right)} = {\frac{p_{i}}{r_{i}\quad s_{i + 1}}\quad \begin{pmatrix}{{r_{i}\quad \frac{{Y_{i + 1}\quad a_{i + 1}^{\prime}\quad g_{i + 1}^{2}} - {X_{i}\quad a_{i}^{\prime}\quad g_{i}^{2}}}{2p_{i}}} - {s_{i}\quad \frac{{Y_{i}\quad a_{i}^{\prime}\quad g_{i}^{2}} - {X_{i - 1}\quad a_{i - 1}^{\prime}\quad g_{i - 1}^{2}}}{2p_{i - 1}}} -} \\{{\frac{\left( {r_{i} - s_{i}} \right)\quad a_{i}^{''}}{2}\quad g_{i}^{2}} - {r_{i}\quad \frac{\left( {{s_{i + 1}\quad g_{i + 1}} - {r_{i}\quad g_{i}}} \right)\quad \left( {{Y_{i + 1}\quad g_{i + 1}} - {X_{i}\quad g_{i}}} \right)}{p_{i}^{2}}} +} \\{s_{i}\quad \frac{\left( {{s_{i}\quad g_{i}} - {r_{i - 1}\quad g_{i - 1}}} \right)\quad \left( {{Y_{i}\quad g_{i}} - {X_{i - 1}\quad g_{i - 1}}} \right)}{p_{i - 1}^{2}}}\end{pmatrix}}$

In both cases the development of two or more deflector arrangements (formonochromatic light) can proceed along the lines described previouslyfor the reflective case, since the dependence on the type of deflectiononly arises within the terms E_(i,1), etc. as described above, and inthe determination of a_(i,1+1) from d_(i,1+1). For example, withrefractive surfaces:$a_{i} = {\arctan \quad \left( \frac{{N\quad \cos \quad \left( d_{i - 1} \right)} - {\cos \quad \left( d_{i} \right)}}{{\sin \quad \left( d_{i} \right)} - {N\quad \sin \quad \left( d_{i - 1} \right)}} \right)}$

It is worth noting that in contrast to the reflective case there may becombinations of d_(i−1,t)and d_(i,t) which cannot be achieved by anyvalues of a_(i,j). The constraints are (for refraction):${{{d_{i} - d_{i - 1}}} \leq {\frac{\pi}{2} - {\arcsin \quad \left( \frac{1}{N} \right)}}}\quad$${{{if}\quad N} > {1\quad {or}\quad {{d_{i} - d_{i - 1}}}} \leq {\frac{\pi}{2} - {\arcsin \quad (N)\quad {if}\quad N}} < 1}\quad$

When considering refractive or diffractive devices it may also behelpful to include within the formulae the impact of a change in λ, thewavelength of light. For refractive devices this involves introducingsome power series expansion describing the dependency of N (the ratio ofthe entry to exit refractive indices) on λ. The degrees of freedom thatarise when adding more deflectors can then be used to reduce thedependence of G(h) on λ instead of (or in conjunction with) achievinghigher order aplanatism.

Examples of two-mirror devices according to the invention produced byfollowing the above design steps are illustrated in FIG. 1 and also inFIGS. 2, 3, and 4 for different pairs of values of parameters k and q.FIG. 1a illustrates how the mirror surfaces of such a device 1 aredefined by rotation of corresponding curves 11 a, 12 a as explainedabove, the curves being shown graphically. The image plane 13 a is shownin the plane of y-axis and orthogonal to the x-axis. FIG. 1a has beenseeded with the parameters a₃=90°, d₂(0)=−90°, p₂(0)=0.25, M₃(0)=(0,0)and M₁(0)=(−4,−1), M₀(0)=(−10⁹,0), B=1/p₁ and Z=+1, and spans about 96%of the complete angle span possible. The resultant device 1, comprisingmirrors 11 b, 12 b together with in a preferred embodiment a planemirror 13 b in the image plane is shown in perspective view in FIG. 1b.The device 1 provides a largely complete angle span onto the image. Itis relatively straightforward to show that the devices illustrated alsosatisfy the sine criterion and thus all the requirements described abovefor first order aplanatism. In these examples, in contrast to thedescription in Siemens-Reiniger-Werke Aktiengesellschaft, the iterationhas been extended to wholly oblique angles.

It should be noted that only within certain ranges of k and q willneither the edge of the inner mirror furthest from the image nor theedge of the outer mirror nearest to the image occlude rays that wouldotherwise strike the image tangentially. All three of the devicesillustrated in FIGS. 2, 3, and 4 are within these bounds (FIG. 2 onlyjust so). FIGS. 3 and 4 have been chosen so that the two mirrors joinup, i.e.:

m _(2,y)(0)=−m _(i,y)(0)→(m _(2,x)(0)² m _(2,y)(0)^(1/2)1

In general the positioning of the mirrors [11 b, 12 b for device 1; 21,22 for device 2; 31, 32 for device 3; 41, 42 for device 4 does notappear to be conveniently describable by analytic formulae, although inthe special case where the device is symmetric and B=1 the cross-sectionconsists of two equally sized confocal truncated ellipses as shown incross-section in FIG. 5.

Light from a far away source subtending a sufficiently small angle ontoa device constructed as shown in FIG. 2 would, with idealised minors, beconcentrated to about 98% of the thermodynamic upper limit. Thecorresponding concentrations for FIG. 3 and FIG. 4 are about 93% and 50%respectively. These figures can be calculated as the proportion ofaperture area through which light passes and is deflected onto theimage, i.e. 1−m_(1,y)(0)²/b². It appears that if the constraint isimposed that the two mirrors join up then however small k is noocclusion of rays takes place (for an object that subtends asufficiently small angle onto the device), i.e. with idealised mirrorsit would be possible to approach arbitrarily dose to the thermodynamicupper limit for a small source. The applicant has tested this down tok=0.05, i.e. equivalent to 99.7% of the thermodynamic upper limit. Atthese levels of thermodynamic efficiency the impact of aberrations orthe imperfections of real-life mirrors would dominate the losses due tothe very slightly incomplete span of angles onto the image plane.However the device would be highly elongated (aperture width 2 butlength from aperture to image plane of circa 11.3).

The impact of aberration, whilst noticeable, should usually bemanageable as long as the source subtends a relatively small angle.Taking solar energy uses as an example, the sun subtends an angle ofabout 0.5° at the surface of the earth. An analysis of the aberrationcharacteristics using the sorts of techniques described above indicatesthat the proportion of light falling other than in the region that theimage of the sun would form in the aplanatic ideal is at most only a fewpercent, at least for the device shown in FIG. 3. Note that from thepreceding analysis above, the degree of aberration G(h) arising from thetwo mirror arrangements is such that the aberration shifts where the raystrikes the image plane by the same amount and in the same direction forboth positive and negative h. Therefore, a particular point at the edgeof the sun would no longer produce a point image but would produce onethat looks approximately like a line, with the line extending somewhatfurther away from the centre of the image than the thermodynaic idealwould require. If desired, it would be possible to get closer to thethermodynamic ideal by shifting M₃(t) in the opposite direction ratherthan having it fixed at the origin, but still rotating the resultingcross-sections around the x-axis rather than using more mirrors toachieve higher order aplanatism.

The size of the object makes no practical difference to the degree ofaberration for solar energy purposes, and for this purpose largeaperture area, e.g. at least 1 m² is likely to be desirable (on thesurface of the earth this would create an image of approximately 0.25cm² in area). For high resolution optics, control of aberration becomesmuch more important, and it may be desirable to make the device as largeas possible to reduce the impact that aberrations might have.

The ultra-high resolution aspects of the invention arise because thereis an inherent link between the ability to achieve ultra-high resolutionin the manner described below and the presence of an ultra-highnumerical aperture imaging layout. It has traditionally been consideredimpossible to circumvent the inherent resolution limits associated withdiffraction. In fact, these limits only actually apply to “far field”devices, see e.g. T. Ito & S. Okazaki, “Pushing the limits oflithography”, Nature, 406, 1027-1031 (2000). This is because it ispossible to achieve higher resolution using the optical equivalent of ascanning funnelling microscope or equivalent lithographic device (i.e. ascanning near field optical microscope or SNOM) or by proximity masklithography, in which the lithographic mask is placed directly on top ofthe surface onto which the image is being projected. However, neither ofthese is ideal from the perspective of manufacture of small-scalesemiconductor architecture. The SNOM can only create an image one pointat a time (or maybe a few points at a time if several SNOM's are linkedtogether), whilst proximity mask lithography requires the mask to bemade to the same accuracy as is intended for the image itself, whichwould present considerable practical difficulties.

Using the present invention, it is possible achieve the same goalswithout the same difficulties using an ultra-high numerical apertureimaging device and some other components. The following explanationconcentrates on the methodology as applied to electromagnetic waves,although the same principles apply to other sorts of waves, e.g. soundwaves or electron waves in electron optics.

The solution to Maxwell's equations caused by a (magnetic) dipoledisturbance in a vacuum radiating from a point on a perfectly conductingplane mirror exactly satisfies the boundary conditions applicable tosuch a mirror if the direction of the dipole is in the mirror plane.Therefore the exact solution to the wave equation from such a dipolesource in the presence of such a mirror is given by the dipole solution,which radiates with a (hemiyspherical wavefront away from the dipolesource (although not one that has a uniform amplitude or polarisationacross the entire wavefront).

Maxwell's equations are time reversible. This means that if adisturbance is arranged to occur on a hemiyspherical shell (the focalpoint of which lies on the plane mirror), the disturbance having thesame spatial distribution of electric and magnetic fields as anoutwardly radiating dipole positioned at the centre of that shell wouldgenerate, but with the direction of either the electric or the magneticfield reversed then the resulting exact solution to Maxwell's equationswould be the corresponding imploding dipole solution, at least up to thetime that the wavefront was arbitrarily close to the centre of thehemisphere.

Suppose that the perfectly conducting plane mirror is replaced with onethat contains some holes, the size of each being small compared to thedistance between it and its nearest neighbor. Suppose also that theboundary conditions on the surface surrounding the mirror are arrangedto be a superposition of the boundary conditions required to generateimploding dipole disturbances as above, each dipole imploding to thecentre of a different hole. Then, as Maxwell's equations are linear, theresulting exact solution to Maxwell's equations is the superposition ofthe solutions arising from each individual imploding dipole disturbance.However, because the electric and magnetic fields of dipole solutionsincrease rapidly (indeed in principle become infinite) at their centre,essentially the whole of any light flux passing through the mirrorarising from a specific imploding dipole disturbance does so in thisinstance through the hole on which it is centred, with essentially noneof the flux going through any other hole.

At least in this limiting case, there is thus a one-to-onecorrespondence between component parts of the superposition ofwavefronts at the boundary and the points through which the flux passesthrough the mirror, and therefore the points at which an image would berecorded were an image recorder or other photosensitive material to bepositioned a small fraction of a wavelength the other side of the planemirror. Note that this result is independent of the wavelength of thelight involved, and hence is not subject to any diffraction basedlimitation as would normally be applicable to imaging systems, althoughit is necessary to create a fine mesh of holes to achieve the desiredresult and the design would therefore be constrained by the opticalproperties of conductors available in the real world. An alternative tocreating a fine mesh of holes would be to use a mirror that wassufficiently thin to allow some light through, as this can be thought ofas a limiting example of the same approach but with the holes beingspread uniformly across the conductor surface.

To create an image which is not diffraction limited it is then necessaryonly to identify some means of achieving an equivalent one-to-onecorrespondence between component parts of the superposition surroundingthe mirror next to the image and points in the object.

This will necessarily involve the use of an ultra-high numericalaperture device, since the approach needs to be imaging in thegeometrical optical sense (to cope with the situation where thewavelength tends to zero) and it needs to span the full or nearly fullrange of angles onto the image plane, in order for it to be possible toapproximate the boundary conditions needed to create an imploding dipolesolution around a complete hemisphere surrounding the centre of theimage plane.

The simplest approach is to use a large, highly elongated, symmetric,double, equally sized confocal truncated ellipsoidal mirror arrangement,as shown in cross-section in FIG. 5, in conjunction with a plane mirror53 placed at the far end of the ellipsoidal mirrors 51, 52 from therelevant light source and a plane mirror 54 at the near end of thesemirrors and for the plane mirors 53 and 54 both to contain a suitablepattern of small holes. If the holes are small enough then the lightsource transmitted though the plane mirror 54 will consist of asuperposition of outwardly radiating (magnetic) dipoles, one for eachhole and if the ellipsoidal mirrors 51, 52 are sufficiently large andsufficiently elongated then for each such dipole the outwardly radiatingwavefronts are deflected in such a manner by the ellipsoidal mirrors tocreate the boundary conditions required to create inwardly-radiatingdipoles as the wavefronts approach the plane mirror 53 next to theimage.

However, such a double truncated ellipsoidal mirror arrangement 5provides no magnification. It therefore suffers from the same practicaldifficulties as a proximity mask lithographic device. Better would be touse an ultra-high numerical aperture imaging system that spans nearlythe complete range of angles onto the mirror next to the image and whichprovides some magnification. This is possible to the level of firstorder aplanatism using just two mirrors. FIG. 1b illustrates one suchdevice whose mirror surfaces are defined by rotation of correspondingcurves as explained above. The device as in FIG. 1b provides an anglespan onto the image to about 96% of the complete angle span possible.Stubbier versions would span a smaller proportion of the complete anglespan, elongated versions a larger proportion (with arbitrarily elongatedversions spanning an arbitrarily high proportion). As indicated in thepreceding analyses, it would also be possible to construct anarrangement involving additional mirrors or lenses, to achieve higherorders of aplanatism.

As noted previously, one practical use for devices according to thepresent invention is in the field of photolithography. An example devicearranged for this purpose is illustrated in cross-section in FIG. 6. Inthis case the two mirrors have surfaces defined by b=1, k=0.25, q=1.832as for the embodiment shown in FIG. 2, except that in this case theobject plane has been taken as x=−6 rather than far away (x) along thenegative x-axis. The device 6 comprises two mirrors 61, 62 which in thiscase are not joined (so that a discontinuity 63 exists between the two).The mirror 62 abuts the image plane 64 which is as shown in theenlargement of area “A” in FIG. 7. A plane mirror 64 a, partiallytransparent, in this case by having many very small holes 66 (a smallfraction of the imaging wavelength wide, but positioned much furtherapart than they are wide) is positioned in the imaging plane with animage recording element 67 (the device or object on which the image isto be projected) disposed a small fraction of the imaging wavelengthbehind the mirror 64 a. Optionally, the device may include additionalcomponents 68 (as discussed further below) to modify the rays comingfrom the object to more closely approximate the desired imploding dipolesolution.

Correspondingly, for the example of FIG. 6, and as shown in theenlargement in FIG. 8, the object, in this case a photo-lithographicmask 80 is positioned adjacent to a plane mirror 65 a, this mirror alsobeing provided with many very small holes 69 (a small fraction of theimaging wavelength wide, and positioned much further apart than they arewide) at conjugate points to those in the mirror 64 a in the imageplane.

If non-zero magnification is desired it is important to note that whilstthe shape of the wavefronts far away from a mirror in the image planemay be as required to generate an imploding dipole solution, the spatialdistinction (and polarisation) of the resulting electric and magneticvectors along the wavefront will not. Instead it will be necessary toattenuate the light (and to rotate the polarisation, probably afterhaving discarded one of the two perpendicular polarisation components)to a different amount depending on the direction in which that part ofthe wavefront approaches the image plane. However, this is not asimpractical as it may first appear. If the required modifications to theamplitude and polarisation are done far away from the image, object andany caustic in between, then the spatial distribution of the requiredadjustments happens to be the same for each dipole component, beingfocused in a region near the centre of the image plane, and thereforeany such adjustment made to one such dipole component willsimultaneously provide the necessary adjustments to all other suchcomponents.

Consider the electric field generated by a (magnetic) dipole in theobject plane pointing in a direction n parallel to the object plane. Thefollowing analysis uses spherical polar coordinates r=(r, θ, φ) where ris the distance from the centre of the dipole, θ is the angle betweenthe vector r and the axis of symmetry and θ is the angle that theprojection of r onto the object plane makes to some fixed vector in thatplane.

The direction and amplitude of the electric field created by such adipole is given by E=(r×n)f(r,t) where f(r,t) is a function of time andthe size of r but not its direction (and × is the vector productoperator). The spatial distribution of its amplitude is thereforeproportional to |r×n| and the angle that it makes to e_(θ) the unitvector in the direction of increasing θ, is α(r,n), say, whereα(r,n)=arcsin(|(r×n)×e_(θ)|/(|r×n∥e_(θ)|)).

Unadjusted, this will then create an electric field on a hemispheresurrounding the image plane which has direction and amplitude as follows(using spherical polar coordinates r′=(r′, θ′, φ′) where r′ is thedistance from the corresponding point on the image plane to the sourceof the dipole and θ is the angle between the vector r′ and the axis ofsymmetry and φ is the angle that the projection of r′ onto the imageplane makes to the same fixed vector as was used to define φ):

(a) The amplitude at the point r′=(r′, θ′φ) where θ′=arcsin(M .sin(θ)),M defining the magnification provided by the device, is apparentlyproportional to |r×n|{square root over (cos(θ)/cos(θ′))}.

(b) The direction at that same point (as given by the angle that itmakes to e′_(θ), the unit vector in the direction of increasing θ′) isevidently a(r,n).

However, for such a wavefront to create an imploding dipole of therequired form, the amplitude actually needs to be proportional to|r′×n′| where n′ is a vector parallel to the image plane, and to have adirection given by α(r′,n′), where

α′(r′,n′)=arcsin(|(r′×n′)×e _(θ′)|/(|r′×n′∥e _(θ′)|)).

Only in the special case where M is unity is this the case (with n′=n).If M is not unity then to create the imploding dipole it would benecessary to insert into the system three further components. Forexample, as illustrated in FIG. 7, it would be possible to use, say,three hemiyspherical shells 68 in front of the image plane, each withits centre at the centre of the image plane (and relatively far from theimage plane compared to the imaging wavelength), each consisting of lotsof small hexagonal tiles, the tiles having appropriate properties. Thetiles in the outermost shell 68 a may each contain a polarising filterto ensure that the direction of polarisation of the electric fieldimmediately after passing through it makes an angle of α(r,n) to e′_(θ).Such an adjustment may be required to discard one of the two independentpolarisations that would normally be radiated from the object, as itdoes not appear to be possible to arrange for the above requirements tobe satisfied simultaneously by both such polarisations. The tiles in thenext shell in 68 b may optionally contain different amounts of anoptically active substance to rotate the incoming light so that itselectric field no longer makes an angle α(r,n) but makes an angle α(r,n)but makes an angle α(r′,n) to e′₇₄ (but preferably without changing theoptical path length). The tiles in the innermost shell 68 c may containdifferent amounts of a semi-transparent substance so that light fallingon different tiles may be attenuated by the right amount to match thatrequired for an imploding dipole solution. There are lots of variants onthis basic theme, e.g. for far away light it would be simpler to replacethe outermost hemispherical shell 68 a with a plane sheet perpendicularto the axis of symmetry placed between the object and the aperture,since the polarisation direction it needs to accept would be everywherethe same.

The above description has concentrated on the use of such an ultra-highresolution apparatus for purposes such as photolithography. It is alsopossible to use this approach to improve the resolution of telescopes ormicroscopes, although the image formed by applying this method directlywould have features that were only a small fraction of a wavelength insize and might therefore be difficult to read. Attentively, a planemirror with small holes spread out over its surface each, say, severalwavelengths from each other, may be placed in the image plane of anultra high numerical aperture telescope/microscope lens or mirrorarrangement and the image viewed from the other side of the mirror witha more traditional microscope. The resulting image would be diffractionlimited, but its amplitude at points in the final image corresponding toeach hole in the plane mirror would very largely relate to light fromconjugate points in the object, rather than other points near to theseconjugate points but too dose to be effectively resolved by atraditional telescope. Therefore, by observing how these amplitudes varywhen the plane mirror (and hence the holes in the mirror) is moved bysmall fractions of a wavelength in each direction, it would then bepossible to build up a higher resolution picture of the object thanavailable with a traditional diffraction limited telescope/microscope(the downside being that a high proportion of the light falling on thetelescope/microscope would be rejected, producing a fainter resultantimage).

Similar results also apply to scalar waves, e.g. sound waves. Indeedbecause the corresponding (scalar) dipole solution is zero on the imageplane everywhere except at the exact centre of the dipole (if the dipoleis pointing in the right direction), it would appear that the mirrorimmediately in front of the image recorder can be dispensed with.Indeed, even with vector waves such as electromagnetic waves, thismirror may be unnecessary. If we create boundary conditions equivalentto those required for an imploding (magnetic or electric) dipole, thedirection of the dipole being parallel to the image plane, then in theabsence of the plane minor the resulting solution to Maxwell's equationsis not that of an imploding (magnetic or electric) dipole. However, eachCartesian component of the electric and magnetic field is still itself asolution to the scalar wave equation. For the component of the electricfield (for a magnetic dipole) or the magnetic field (for an electricdipole) parallel to the image plane the mathematics seems to be the sameas for the imploding scalar dipole referred to above, and therefore itis possible that it is zero everywhere on the image plane except at theexact centre of the dipole even when no plane mirror is present there.Therefore in this instance it would seem that there is no energy flowexcept at the exact centre of the dipole even when no plane mirror ispresent in the image plane. Such a result seems to be consistent with arecent paper by J. B. Pendry “Negative Refraction Makes a Perfect Lens”,Physical Review Letters 85, No 8 3966-3969 (2000). The author of thatpaper claims that it is possible to produce a diffraction free lensusing a lens made of material with a negative refractive index. Hispaper demonstrates that, with this unusual type of lens, evanescentwaves that normally disappear when light passes through a lensarrangement reappear in the wavefront as it approaches the origin, whichis similar to the wavefront that the apparatus disclosed here shouldcreate. Thus if Pendry's claim is correct then the apparatus disclosedhere should also be diffraction free even without a partiallytransparent plane mirror in the image plane.

What is claimed is:
 1. A method of designing a two-mirror high numericalaperture imaging device comprising the steps of: (a) determining theposition of each consecutive point of a cross-section through the x-axisof a first mirror located nearest an object for imaging in use and asecond mirror located nearest the image in use, iteratively for across-section in the plane z=0, where the x and y coordinates of eachsuccessive point on the first and second mirrors in the cross-sectionare determined by the functions M₁(t)≡(m_(i,x)(t), m_(1,y) (t),0) andM₂(t)≡(m_(2,x)(t), m_(2,y)(t),0) respectively, t being an iterationcounter; where the functions M₀(t)≡(m_(0,x)(t), m_(0,y)(t),0)≡(f,0,0)and M₃(t)≡(m_(3,x)(t), m_(3,y)(t),0)≡(0,0,0) are set to define thecentres of an object plane and an image plane, respectively; where a₁(t)and a₂(t) are the angles that tangents to the first and second mirrorsmake to the z-axis, and a₀(t) and a₃(t) are the angles that the objectand image planes make to the x-axis (being 90° for all t for the mirrorsto be rotationally symmetric about the x-axis); d_(i)(t) (for i=0, 1, 2)is the angle that a ray from M_(i)(t) to M_(i+1)(t) makes to the x-axisand p_(i)(t) (for i=0, 1, 2) is the distance between M_(i)(t) andM_(i+1)(t), so that:${d_{i}\quad (t)} = {\arctan \quad \left( \frac{{m_{{i + 1},y}\quad (t)} - {m_{i,y}\quad (t)}}{{m_{{i + 1},x}\quad (t)} - {m_{i,x}\quad (t)}} \right)}$${a_{i}\quad (t)} = {\frac{d_{i}\quad (t)}{2} + \frac{d_{i - 1}\quad (t)}{2}}$${p_{i}\quad (t)} = \sqrt{\left( {{m_{{i + 1},x}\quad (t)} - {m_{i,x}\quad (t)}} \right)^{2} + \left( {{m_{{i + 1},y}\quad (t)} - {m_{i,y}\quad (t)}} \right)^{2}}$

(b) selecting suitable parameters to define the size and shape of thedevice—for use with a far away source (of the order of f=−10⁹ distantalong the negative x-axis) define the overall width of the device by thevalue of B, B=b/p₀(0); define two further parameters m_(2,y)(0)=k andm_(2,x)(0)=−q, and taking m_(1,x)(0)=m_(2,x)(0)−δ, where δ is small or 0(thereby defining one limit of the acceptable range of iterated values);(c) assigning values to m_(1,y)(0) and Z:$Z = {{1\quad m_{1,y}\quad (0)} = {- \frac{m_{2,y}\quad (0)}{\left( {{m_{2,x}\quad (0)^{2}} + {m_{2,y}\quad (0)^{2}}} \right)^{1/2}}}}$

(d) updating the values of M₁(t) and M₂(t) according to the followingiterative formulae for a small value of h, where w_(i)(t) is a functioncorresponding to the distance between M_(i)(t) and M_(i)(t+1) andr_(i−1)(t)=sin(a⁻¹(t)−d_(i−1)(t)) s_(i)(t)=sin (a_(i)(t)−d_(i−1)(t))${{M_{i}\quad \left( {t + 1} \right)} \equiv \begin{pmatrix}{m_{i,x}\quad \left( {t + 1} \right)} \\{m_{i,y}\quad \left( {t + 1} \right)}\end{pmatrix}} = {{M_{i}\quad (t)} + {w_{i}\quad \begin{pmatrix}{\cos \quad \left( {a_{i}\quad (t)} \right)} \\{\sin \quad \left( {a_{i}\quad (t)} \right)}\end{pmatrix}\quad h}}$

 where the distance parameters w_(i)(t) for w₁, w₂, a₁′w₁ and a₂′w₂satisfy the four simultaneous equations: w ₂ h=E ₁(w ₁ h,0)0=E ₂(w ₂ h,w₁ h) w ₂ h=E ₁(0,h) ZBh=E ₂(w ₂ h,0)  (h corresponding to the distanceaway from the centre that rays eventually strike the image plane adistance ZBh from its centre and a_(i)′(t)g_(i) is the differencebetween the angle that the i′th mirror makes to the x-axis at pointM_(i)(t) and the angle that it makes at a distance g_(i) from M_(i)(t),for a smooth surface this being a linear function of g_(i)for smallg_(i), a_(i)′(t) being the derivative of a_(i)(t) with respect to t);w₁(t) and w₂(t) further satisfying the relationships:${w_{2}\quad h} = {{{E_{1}\quad \left( {0,h} \right)\quad \frac{p_{1}\quad r_{0}\quad h}{p_{0}\quad s_{2}}\quad 0} - {ZBh}} = {\left( {{E_{2}\quad \left( {{w_{2}\quad h},{w_{1}\quad h}} \right)} - {E_{2}\quad \left( {{w_{2}\quad h},0} \right)}} \right) = {\frac{p_{2}\quad r_{1}}{p_{1}\quad s_{3}}\quad w_{1}\quad h}}}$

 where E_(i)(g_(i), g_(i−1)) represents the distance from M_(i+1)(t)that a ray will strike the i+1′th surface if it comes from a point onthe i−1′th surface that is a distance g_(i−1) from M_(i−1)(t) via apoint on the i′th surface that is a distance g_(i) from M_(i)(t), forsmall g_(i−1) and g_(i), such that for deflection by reflection:${E_{i}\quad \left( {g_{i},g_{i - 1}} \right)} = {{\left( {\frac{r_{i}}{s_{i + 1}} - \frac{p_{i}\quad s_{i}}{p_{i - 1}\quad s_{i + 1}}} \right)\quad g_{i}} + {\left( \frac{p_{i}\quad r_{i - 1}}{p_{i - 1}\quad s_{i + 1}} \right)\quad g_{i - 1}} + {\left( \frac{2p_{i}}{s_{i + 1}} \right)\quad a_{i}^{\prime}\quad g_{i}}}$${i.e.\quad w_{2}} = {{\frac{p_{1}\quad r_{0}}{p_{0}\quad s_{2}}\quad w_{1}} = {{- {ZB}}\quad \frac{p_{1}\quad s_{3}}{p_{2}\quad r_{1}}}}$

(e) ending the iteration when m_(2,x)(0) approaches zero therebydefining the other limit of the acceptable range of iterated values; (f)rotating the curves thus produced, which define the shapes of the twomirrors, around the x-axis to define a complete, three-dimensionaltwo-mirror arrangement.
 2. A high numerical aperture imaging devicecomprising at least two mirrors made according to the method of claim 1.3. A high numerical aperture imaging device (1) according to claim 2comprising first and second axially-symmetric curved mirrors (11 b,12 b)for focussing the image of an object onto an image point (13 a), whereinthe first and second curved mirrors (11 b,12 b) are arranged toeffectively create inwardly imploding dipole-like solutions to theapplicable wave equation, to concentrate the energy flux arriving at theimage plane (13 a) from a given point in the object more than would bepossible were the image formation to be subject to the diffractionlimits that generally apply to far field devices.
 4. A device accordingto claim 3 further comprising a wave attenuation element (68 c) and awave polarisation-rotating element (68 a) designed so that the spatialdistribution of the amplitude and polarisation of a wavefront as itapproaches the image plane is rendered more closely consistent with thatrequired to generate dipole-like solutions to the wave equation.
 5. Ahigh numerical aperture imaging device (1) according to claim 2 furthercomprising a partially transparent plane mirror (13 b) positionedproximate to the image plane.
 6. A device according to claim 5 furthercomprising a wave attenuation element (68 c) and a wavepolarisation-rotating element (68 a) designed so that the spatialdistribution of the amplitude and polarisation of a wavefront as itapproaches the image plane is rendered more closely consistent with thatrequired to generate dipole-like solutions to the wave equation.
 7. Adevice (6) according to claim 3 adapted to produce highly accuratelithographic images for use in semiconductor/microchip manufacture.
 8. Adevice according to claim 2 adapted to concentrate waves satisfyingequivalent ‘ballistic’ equations of motion.
 9. A device according toclaim 2 adapted to project waves satisfying equivalent ‘ballistic’equations of motion.
 10. A device according to claim 2 adapted toconcentrate physical entities other than waves which satisfy equivalent‘ballistic’ equations of motion.
 11. A device according to claim 2adapted to project physical entities other than waves which satisfyequivalent ‘ballistic’ equations of motion.
 12. A device according toclaim 2, wherein the mirrors retain their required shape by being partof an inflated structure.
 13. A device according to claim 2, wherein themirrors retain their required shape by being rotated at a sufficientlyrapid speed.
 14. A device according to claim 12 adapted to concentratesunlight to a high temperature for the purpose of generating electricpower.
 15. A device according to claim 12 in which concentrated sunlightis used to create direct thrust for powered flight by evaporation of apropellant.
 16. A device according to claim 13 adapted to concentratesunlight to a high temperature for the purpose of generating electricpower.
 17. A device according to claim 13 in which concentrated sunlightis used to create direct thrust for powered flight by evaporation of apropellant.
 18. A device according to claim 2 for interlinking ofoptical networking components, the device further comprising a solidstate optical emitter in the source/image plane.
 19. A device accordingto claim 2 for interlinking of optical networking components, the devicefurther comprising a solid state detector in the source/image plane. 20.A device according to claim 2 in which the first and second mirrors arejoined, such that: m_(2,y)(0)=−m_(1,y)(0)→(m _(2,x)(0)² +m_(2,y)(0)²)^(1/2)=1.
 21. A device according to claim 2 in which theshape of the first and second mirrors is further modified to compensatefor higher order aberrations by deviating from the aplanatic ideal insuch a way that rays coming from a circular or far away spherical objectfall primarily within a circular shaped image, the circle being smallerthan the shape of the image that would be formed by the correspondingideal aplanatic arrangement.
 22. A high numerical aperture deviceaccording to claim 2 further comprising at least one additional surfaceadapted to exhibit aplanatism of order greater than one.
 23. A highnumerical aperture device according to claim 3 further comprising atleast one additional surface adapted to exhibit aplanatism of ordergreater than one.
 24. A high numerical aperture device according toclaim 4 further comprising at least one additional surface adapted toexhibit aplanatism of order greater than one.
 25. A high numericalaperture device according to claim 5 further comprising at least oneadditional surface adapted to exhibit aplanatism of order greater thanone.
 26. A high numerical aperture device according to claim 6 furthercomprising at least one additional surface adapted to exhibit aplanatismof order greater than one.
 27. A high numerical aperture deviceaccording to claim 7 further comprising at least one additional surfaceadapted to exhibit aplanatism of order greater than one.